CC6+ Unit 7 Equation and Inequality
Notes 7.1 Expressions, Equations and Inequalities
I Expression :
_ a mathematical phrase containing numbers, operations ( ), and/or variables
‐ Expressions can be evaluated or simplified but they cannot be "solved"
Examples:
12
9 ÷ 3 + 42
+ 4(3)
5𝑥 − 2
9𝑚 − 12
3+𝑥+2
4
3−2+4
5𝑤 − 3𝑤
4+𝑦+𝑦+𝑦
5𝑦 + 𝑦
Numerical expressions can be simplified to one number.
9 ÷ 3 + 42
12
4
3−2 + 4
+ 4(3)
Algebraic expressions can be simplified
5𝑦 + 𝑦
4+𝑦+𝑦+𝑦
5𝑤 − 3𝑤
Algebraic expressions can be evaluated for given variables
5𝑥 − 2
when 𝑥 = 3
9𝑚 − 12 when 𝑚 = 4
II Equation
‐ Equations can be solved and usually have one solution
‐ Like a balanced scale
‐ a mathematical sentence that states two expressions are equivalent
‐ has an equal sign in between
Examples:
3+𝑥+2=9
5𝑤 − 3𝑤 = 9 ÷ 3 + 22 − 1
12
3 + 5 = 5𝑥 − 2
+ 4(3) = 9𝑚 − 12
4
III. Inequality
‐ a mathematical sentence that states one expressions is greater than or less than
another expression
-has inequality signs
‐ Inequalities can be solved but have many solutions
‐ Like an unbalanced scale
Examples :
12
3 + 𝑥 + 2 ≤ 9 5𝑤 − 3𝑤 ≥ 9 ÷ 3 + 22 − 1 3 + 5 < 5𝑥 − 2
+ 4(3) > 9𝑚 − 12
4
Video: http://www.youtube.com/watch?v=jQAxpFAKQ6M&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Grade 6
Name: ______________________
Date: _______
Period: __
Notes 7.1 Expressions, Equations and Inequalities Guided Notes
3y2 + 2y – 3 = 30
5n – (8 ÷ 2)
7m – m3 > 6
____________________
List characteristics of each “item” above and then create a Venn diagram to compare and
contrast similarities and differences.
Video: http://www.youtube.com/watch?v=jQAxpFAKQ6M&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.1 C
I.
Categorize each of the following mathematical “items” into the proper columns:
5x + 3x + 2x
t–7=7
45
45x – 4 = 86
2w + 3 = 23
Expressions
4 + 9y > 36
98 ÷ 2x ≤ 14
u – 5 ≥ 12
Equations
5 – 3 + 12 x 2
5d – d + b
9 < 23b
7g =
10
2𝑔
Inequalities
II.
Write each “statement” mathematically and identify it as an expression, equation, or
inequality:
1. Each fountainhead had two nozzles each and there were 5 faucets.
__________________________________________________________________________.
2. All three of the fountainheads sprayed out the same amount of water, which totaled twelve
thousand gallons of water.
__________________________________________________________________________.
3. The two filters processed at least five-hundred-thousand gallons of water a week.
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.1H
Identify Expressions, Equations, and Inequality
Name___________________________
Date_____________ Pd____
Determine if each statement/item applies to Expressions, Equations, and/or Inequalities. Some statements may
apply to more than one. Put a check in the correct column(s).
Expression Equation
Inequality
1) Contains an equal sign
2) Cannot be ‘solved’
3) 𝟑𝒙 + 𝟒
4) 𝟏𝟎 < 21
5) Has multiple solutions
6) 𝟏𝟐 = 𝟑 + 𝒙
7) Can contain numbers
8) A mathematical sentence
9) 𝟕𝒘
10) 𝟒 = 𝟏𝟐 ÷ 𝟑
11) Can contain variables
12) Can be compared to a scale
13) A mathematical phrase
14) 𝒙 ≥ 𝟗
15) 𝒙 + 𝟖 < 37 − 2
16) Like a balanced scale
17) 𝟕 + 𝟏
18) 𝟕 + 𝟏 = 𝟖
19) 𝟕 + 𝟏 ≥ 8
20) Can contain operation symbols (+,-,x,÷)
𝟑
21) 𝟒 + 𝟕(𝟐) − 𝟏𝟐 + 𝟑𝟎
22) 𝟓 − 𝒙 = 𝟏𝟐
23) 𝟓 − 𝒙
24) Usually has one solution
25) Like an unbalanced scale
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.1 C
Equations VS inequality
Sorting Cards
3 + 5 = 5𝑥 − 2
5𝑤 − 3𝑤 = 9 ÷ 3 + 22 − 1
12
+ 4(3) > 9𝑚 − 12
4
3+𝑥+2<9
5𝑥 − 2
7+9
3 + 5 ≤ 5𝑥 − 2
12
+ 4(3)
4
3+𝑥+2=9
5𝑤 − 3𝑤
9 ÷ 3 + 22 − 1
9
Expressions VS Equations and Inequalities
Sorting cards
3 + 5 = 5𝑥 − 2
5𝑤 − 3𝑤 = 9 ÷ 3 + 22 − 1
12
+ 4(3) > 9𝑚 − 12
4
3+𝑥+2<9
5𝑥 − 2
7+9
3 + 5 ≤ 5𝑥 − 2
12
+ 4(3)
4
3+𝑥+2=9
5𝑤 − 3𝑤
9 ÷ 3 + 22 − 1
9
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
IDENTIFYING EXPRESSIONS, EQUATIONS, & INEQUALITIES
Sample Quiz Unit 7 Part 1
Problems
Answers
Identify each item as an expression, equation, or inequality. Write your answer in the answer column.
1. 7𝑤 + 8 + 6𝑤 − 3
2.
20𝑤
4
> 10
3. 4𝑥 = 8
4. 14 + 9
5. 18𝑔 ÷ 2 ∙ 3 = 5𝑔 − 6
6. 𝑥 ≤ 4
7. 7
8. 8 = 8
9. 9 < 10
10. 𝑎 + 𝑏 − 𝑐𝑑 > 𝑐𝑑 + 𝑒𝑓𝑔
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Notes 7.2A
Solutions to Equations and Inequalities Guided Notes
SOLUTION? It’s the value or values that make an equation or inequality true.
EXAMPLE: Is t = ____ a solution to the following ________________________?
6t + 13 = 6t – 13
14 – 2t = 2t – 14
9t ÷ 9 = 4t ÷ 4
STEP 1: _____________________ – “plug-in” or replace the variable with the possible
solution.
____________________
__________________ ________________
STEP 2: _____________________ – evaluate the __________________________ on
either side of the equation/inequality.
_____________________
________________
____________
____________
________________
____________
STEP 3: ___________________ – determine if the mathematical statement is _________.
__________________ ______________________
__________________
Is t=____ a solution for the following ____________________________?
(* Follow the same steps as above for inequalities as equations.)
9t ÷ 9 < 4t ÷ 4
_____________
9t ÷ 9 > 4t ÷ 4
_____________
9t ÷ 9 ≤ 4t ÷ 4
9t ÷ 9 ≥ 4t ÷ 4
_______________
______________
______________
______________
_______________
______________
______________
_______________
______________
______________
________________
____________
______________
______________
Video :
http://www.youtube.com/watch?v=0EMUtIV13H8&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41lPFTM
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Notes 7.2 B
Solutions to Equations and Inequalities
Solution the value or values that make an equation or inequality true
Is 𝒎 = 𝟒 a solution to 𝟓𝒎 + 𝟏𝟎 ≤ 𝟕𝒎 − 𝟐 ?
To determine if a given value is a solution:
1. Substitute the given value into the equation or inequality
2. Simplify the expression on either side of the equation or inequality
3. Determine if the simplified expressions satisfy the equal sign or inequality symbol
𝟓𝒎 + 𝟏𝟎 > 𝟕𝒎 − 𝟐
5(4) + 10 > 7(4) − 2
Simplify the expression on either side of the equation or inequality
NOTE: the >,<, or = sign seperates the 2 sides (2 expressions
𝟓𝒎 + 𝟏𝟎 > 𝟕𝒎 − 𝟐
5(4) + 10 > 7(4) − 2
𝟐𝟎 + 𝟏𝟎 > 𝟐𝟖 − 𝟐
𝟑𝟎 > 𝟐𝟔
Determine if the simplified expressions satisfy the equal sign or inequality symbol.
http://www.youtube.com/watch?v=MG4DvlWQaGE&feature=share&list=PLNDkuWRw1gGTga
Yk6dQhGp10UT41lPFTM
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7-2H State whether the given value is a solution to the equation or inequality.
Write YES or NO. Show work to prove your answer.
1) 2𝑥 + 7 = 17;
𝑥=5
8)
5𝑤 < 3𝑤 + 6 ;
2) 35 = 7ℎ − 8 ;
ℎ=6
9) 63 < 3 + 6𝑟 ;
3) 63 ≤ 3 + 6𝑟 ;
𝑟 = 10
10) 50 − 3𝑥 = 42 ;
4) 9 + 6𝑎 = 57 ;
𝑎=8
5) 10 + 4 > 20 − 3𝑣 ;
6) 9ℎ = 20 + 6ℎ ;
ℎ=7
7) 8 + 6𝑐 ≤ 9𝑐 − 30 ;
CC6+ Unit 7
𝑣=4
𝑐=4
𝑤=2
𝑟 = 10
11) 9𝑘 + 3 < 8 ∙ 3 ;
12) 5𝑔 − 15 ≥ 60 ;
13)
𝑥 = 12
𝑘=2
𝑔 = 15
79 − 8𝑝 = 34 + 𝑝 ;
14) 8𝑟 + 1 = 5𝑟 + 10;
𝑝=5
𝑟=3
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CC6+ Unit 7 Equation and Inequality
Practice 7-2 H2 State whether the given value is a solution to the equation or inequality. Write YES or NO.
CHALLENGE: If the value is not a solution, can you determine which value(s) would be a solution?
1 . 5𝑥 – 8 = 18 + 4, for 𝑥 = 6
2. 4𝑥 2 – 5(5) = 12, for 𝑥 = 3
3. (8 – 𝑛)2 + 13 ≥ 23 , for 𝑛 = 3
4. 17𝑥 − 8(2𝑥 − 4) > 32, for 𝑥 = 3
5. 2𝑥 + 12 + 8𝑥 > 32, for 𝑥 = 2
6. 6(3𝑥 − 2) + 5 < 50, for 𝑥 = 3
Test each value in the ‘Replacement Set’ column to determine if the values are solution(s) to the given
equation/inequality. Be sure to list all numbers that work to make the statement true. There may be 1, more than
one, or no solutions to each. ** Do your work in the space below the chart.
Equation
Replacement Set
Solution(s)
7. 5𝑥 + 2 = 17
{1, 2, 3, 4}
8.
3𝑥 − 2 > 4
{2, 3, 4, 5}
9.
2𝑥 2 + 4 = 54
{1, 3, 5, 7}
10. 7𝑥 − 7 < 30
{2, 4, 6, 8}
11. 2(2𝑥 + 4) > 20
{3, 5, 6, 9}
12. 5𝑥 – 6 = 24
{1, 2, 3, 4}
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Solving 1-Step Equations and Inequalities Guided Notes
How do we find the value of the variables in these equations?
______________________, where we just guess different values until the math statement is true?
What do equations and a balanced scale have in common?
The ________________ on both sides of the equal/inequality sign must have the __________.
Therefore, whatever operation you do to one side of the equation, you must do the same operation to the
other side as well.
If we can ________________ the variable on side of the equation, we can solve for the variable!
_________________ the variable is a strategy, where the goal is to get the variable all by itself on one side of
the equal or inequality sign.
x–7 =
5
8y
a.
=
24
b.
1st
𝑚
3
=
9
c.
: Ask, what is being done to the variable?
a. ________________________ b. ________________________ c. _____________________
2nd
: Ask, what is the inverse operation? “How do I undo this operation?”
a. __________
b. _________________
c. ________________________
3rd
: Perform this operation to both sides of the equation/inequality to isolate the variable and keep the
equation/inequality balanced.
a. x – 7 = 5
b. 8y = 24
___ = ____
____ = ____
y+8
>
CC6+ Unit 7
𝑚
c. 3( 3 )
____
=
=
3(9)
_____
INEQUALITIES are solved just like equations, but remember that
your final answer is not single, but rather a range of multiple
solutions! 15 > y + 8
-8
-8 ( y must be any number
___
> y
less than 7.)
Page 11
CC6+ Unit 7 Equation and Inequality
Practice 7.3
One –Step Equation
Solve each equation. Make sure to show your work by using the inverse property. Circle your answer.
1. 8 + v = 26
13. 10𝑛 = 40
2. 3 + p = 8
14.
3. 15 + b = 23
15.
𝑣
8
=2
𝑘
11
= 14
4. m + 4 = 12
16. 15𝑥 = 0
5. x – 7 = 13
17. 17x = 204
6. m – 9 = 23
18. 21 = 7n
7. p – 6 = 5
19.
8. v – 15 = 26
20. 126 = 14𝑘
9. n + 9 = 16
21. 143 = 11x
10. 8x =104
22. 16 + x = 19
11. 14b = 56
23.
5 = 18
24.
17 = x – 15
12.
𝑏
18
𝑚
4
= 13
𝑎
=6
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.3
Hands on Equation
Equation:
Check:
Class Example:
Class Example:
Example 1:
Example 2:
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
2x + 3x = 15
Class Example:
4x + 2x = 15 + 3
Example 3:
x + 7x = 2 + 14
Example 4:
x + 4 = 13
Class Example:
5 + x = 12
Example 5:
17 = 9 + x
Example 6:
5x – 2x = 12
Class Example:
8x – 6x = 20
Example 7:
7x – x = 18
Example 8:
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7. 4
Equations &Word Problems with Decimals
Solve and check the equations.
1) 2.23 + x = 6.5
2) x – 4.75 = 9.2
3) 0.06 + y = 3.6
Write an equation for each word problem. Solve and check.
5) A plastic jug holds 5.2l of liquid. A water bottle holds 3.9l less liquid than the jug. What is
the volume of liquid in the water bottle?
6) Sammy had $964.21in his checking account at the beginning of the month. The bank
posted a balance of $2695.36 on the 15th of the month. How much money had Sammy
deposited during this part of the month?
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.4 One step Equation
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.5C
CC6+ Unit 7
One Step Equation
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CC6+ Unit 7 Equation and Inequality
Practice 7.5H Equation Word Problems
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Notes 7.6
Graphing Equations and Inequalities on Number Lines Guided Notes
Graphing an Equation on a Number Line:
(equal)
X=4
The value of x, which is 4, is represented by a darkened _________ at 4 to the ___________ of 0 on the
number line.
Graphing Inequalities on a Number Line:
(greater than)
Y>3
The value of y, which is any number _____________ than 4, is represented by an _______________ at 3 to the
right of zero with an ______________ pointing to right on the number line. An open circle means it
_____________ equal that value, and the arrow (ray) pointing to the right means that it can be any value
______________ than the number that is circled.
Z<5
(less than)
The value of z, which is any number ________ than 5, is represented by an _______________ at 5 to the right
of zero with an _____________ pointing to left on the number line. An open circle means it ____________
equal that value, and the arrow (ray) pointing to the left means that it can be any value __________ than the
number that is circled.
(greater than or equal)
W ≥ -2
The value of w, which is ___________ to -2 and any number ____________ than -2, is represented by a
________ at 2 to the left of zero with an arrow pointing to __________ on the number line. The point means
it _____________ that value, and the arrow (ray) pointing to the right means that it can be any value
_______________ than the number where the point is located as well.
(less than or equal)
V≤1
The value of v, which is ___________ to 1 and any number ___________ than 1, is represented by a
___________ at 1 to the right of zero with an arrow pointing to __________ on the number line. The point
means it ____________ that value, and the arrow (ray) pointing to the left means that it can be any value
_____________ than the number where the point is located as well.
WCPSS Video:
http://www.youtube.com/watch?v=TIdrfYfNmDs&feature=share&list=PLNDkuWRw1gGTgaYk6dQhGp10UT41l
PFTM
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.6 C Graphing Inequalities
Graph Inequalities Inequalities can be graphed on a number line. This helps you to
see which values make the inequality true. You can also write inequalities for a graph.
An open dot indicates that the number marked does not make the sentence true.
A closed dot indicates that the number marked does make the sentence true.
The direction
of the line indicates whether numbers greater than or less than the number
Example
1
marked make the sentence true.
Graph each inequality on a number line.
a. x ≤ 8
6
b.
7
8
9
10
x >8
6
7
8
9
10
The closed dot means 8 does make the sentence true. The line means that numbers less than 8 make the
sentence true.
The open dot means 8 does not make the sentence true. The line means that numbers greater than 8 make
the sentence true.
Write an inequality for each graph.
a.
b.
-5
-4
-3
-2
-1
3
0
The open dot means -2 is not included in the
graph. The arrow points left, so the graph includes
all numbers less than -2.
The inequality is x < -2.
4
5
6
7
8
The closed dot means 5 is included in the graph.
The arrow points right, so the graph includes all
numbers greater than 5.
The inequality is x ≥ 5.
Exercises
Graph each inequality on a number line.
2. a ≤ -2
1. x > 7
5
6
7
8
9
-10 -9
-3
-2
-1
0
-6
5. t ≥ -5
4. w > -9
-11
-4
3. d < -4
-8
-7
-7
-6
-5
-4
-3
-2
6. n < -11
-5
-4
-3
-13
-12
-11
-10
-9
Write the inequality for each graph.
7.
8.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
9. .
-5 -4 -3 -2 -1
0
1
2
3
4
5
10.
-20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10
0
1
2
3
4
5
6
7
8
9
10
Video Graphing Inequality http://www.virtualnerd.com/middle-math/equations-functions/inequalities/number-line-graph-inequality
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Notes 7.6 Inequalities
A mathematical sentence that contains any of the symbols listed below is called an inequality.
<
>
≤
≥
• is less than
• is greater than
• is fewer than
• is more than
• is less than or equal • is greater than or
to
equal to
 at most
 at least
 no more than
 no less than
Example 1 Write an inequality for the sentence.
• is no more than
• is no less than
• exceeds
Fewer than 70 students attended the last dance.
Words Fewer than 70 students attended the last dance.
Symbols Let s = the number of students.
• is at most
• is at least
Inequality s < 70
You can substitute a value for a variable in an inequality and determine whether the value makes the inequality true or
false.
Example 2
For the given value, state whether each inequality is true or false.
a. 5y - 6 < 14; y = 5
b. r - 16 ≥ -12; r = 4
5y - 6 < 14
Write the inequality.
r - 16 ≥ -12
5(5) - 6 < 14 Replace the variable with the given value.
4 - 16 ≥ -12
19 < 14
Simplify.
-12 ≥ -12
This sentence is false.
Although -12 > -12 is false, -12 = -12 is true.
So, this sentence is true.
Exercises
Write an inequality for each sentence
1. The maximum diving depth is no more than 45 feet below sea level.
2. Adult male elephants can weigh over 12,000 pounds.
_________________
_____________________________________
3. The maximum fee for any student is $15.________________________________________
4. You must be at least 38 inches tall to ride the roller coaster._________________________________
For the given value, state whether the inequality is true or false.
5. m + 8 ≥ 5; m = -3
6. 4+ p < -2; p = 6
7. b + 12 ≤ 15; b = 1
8. j + 7 < -8; j = 0
Video : Definition of Inequality http://www.virtualnerd.com/middle-math/equations-functions/inequalities/inequality-definition
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Notes 7.7 Solve Inequalities by Adding or Subtracting
Use the Addition and Subtraction Properties of Inequalities to solve inequalities. When you add or subtract
a number from each side of an inequality, the inequality remains true.
Example
Solve 12 + y > 20. Check your solution.
12 + y > 20
Write the inequality.
12 - 12 + y > 20 - 12
y>8
Subtraction Property of Inequality
Simplify.
To check your solution, try any number greater than 8.
?
12 + y > 20
CHECK
12 + 9
?
20
Write the inequality.
Replace y with 9.
21 > 20 ✔
This statement is true.
Any number greater than 8 will make the statement true. Therefore, the solution is y > 8.
Exercises
Solve each inequality. Check your solution.
1. 12 < 8 + b
2.
t- 5 >4
3.
p + 5 < 13
4. 5 > y - 6
5.
21 < n -18
6.
s-4≤3
7. 14 > w + (-2)
8. j + 6 ≥ -4
10.
𝒃 + 𝟑. 𝟖𝟗 ≤ 𝟒. 𝟐𝟓
11.
1
9. z + (-4) < -2.5
1
g - 24 ≥ 3 4
Video : Learnzillion http://learnzillion.com/lessons/2614-write-and-graph-inequalities-shopping
http://learnzillion.com/lessons/2615-write-and-graph-inequalities-temperature
CC6+ Unit 7
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CC6+ Unit 7 Equation and Inequality
Practice 7.6 One-Step Inequality
CC6+ Unit 7
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